Balancing Acts

Purpose

The unit involves students in solving problems that can be modelled with algebraic equations or expressions. Students are required to describe patterns and relationships using letters to represent variables.

Specific Learning Outcomes
  • to predict further members in patterns of equations using relationships within the equations
  • to develop function rules to describe relationships
  • to find specific values for variables from given relationships
Description of Mathematics

This unit provides novel ways of looking for patterns. At the same time there is very definitely a bias towards algebra. The material gets very close to setting up algebraic equations and so should be a good preparation for Level 5 Algebra.

Links to Numeracy

This unit offers many opportunities for students to develop numeracy understandings. Most of the problem sets in Session Two could be used to focus on aspects of strategy development including different ways numbers can be partitioned when calculating and place value understandings.

Session Three is an ideal context for developing understandings about the reversibility of operations. As students are writing and solving “What’s my Number?” problems encourage them to focus on the ways to “undo” each problem to solve it. Note that the order of operations is important; the last operation listed in the problem needs to be the first operation to be reversed.

How did you solve Jane’s problem? Why did you choose that way
What did you need to do to the numbers first? Second?
Why did you choose to do it that way?
What happens if you change the way you do things? Why?

Once students have a sound understanding of the reversibility of operations encourage them to make the numbers they are using larger. For example, if you take my number, add 256, multiply it by 7 and take away 312 you get 1641. (23 since 23 + 256 X 7 – 312 = 1641). This will encourage the use of more sophisticated mental strategies for number operations.

Required Resource Materials
plastic cups of different colours (available from dollar shops)- red, yellow and blue are referred to below

Copymasters for Session 1

Copymasters for Session 2

counters or coloured cubes

Copymasters for Session 3

1.5 plastic bottles

Copymasters for Session 4

Copymasters for Session 5

Key Vocabulary

 equations, recursion, relationships, pattern, functional rule, input numbers, output numbers

Activity

It might be useful for students to do some parts of each session each day of the unit. This is because the problems in each session build up in complexity and student understanding is probably best developed over several days in each problem type.

Session 1

Make up a set of three containers by trimming down 1.5 litre plastic soft drink bottles. It is preferable that these containers are of the same type. Straight-sided bottles are better than curved ones as they make it easier for the students to predict the relationships. The three cut down containers should hold different amounts of water when full. Make their capacity no more than 500 ml and ensure that these capacities are not multiples of each other, e.g. 120 ml, 230 ml, and 400 ml would be better than 100 ml, 200 ml, and 400 ml.

Label the containers p, q, and r respectively. Use them to set problems for the students by pouring from the p, q, and r containers into the large container.

With each problem students must write an expression for the water level or do some other similar activity. Here are some examples. You can extend these as appropriate for your class.

First we give two examples.


  1. diagram.

  2. diagram.
  1. Pour four lots of p into the 3 litre bottle and pour from it a container of r and a container of q.
    (This gives 4pr q.)
  2. Pour five lots of r into the large bottle and take out two lots of p. ( 5r – 2p)
  3. Pour three lots of r and three lots of q into the large bottle and pour out two lots of p.
    (This gives 3r + 3q – 2p which could be written as 3(r + q) – 2p.
  4. What does 3(r + q) – 2p mean in terms of pouring? (Put an r and a q lot together 3 times and then pour off 2 lots of p.)

    These types of problems allow opportunities for students to discover equivalence of expressions, particularly in the cases where order does or does not make a difference to the result, eg. 2r + p = p + 2r = r + p + r.
  5. Which of these are correct and which are false? Explain your answers.
    1. 3(r + q) – 2p is the same a q + r + 2(rp) + q ;
    2. 4prq is the same as p + 2(prq) + q + r + p ;
    3. 2r + p + q = 2(p + q) + 3rqp – 2r .

Session 2

Students look for patterns within each equation set and use these patterns to predict further equations in the set. They may do this using recursion, that is finding a relation between consecutive equations, rather than by looking for relationships within the equations themselves. Highlight relationships that might be found between the numbers in each set of equations and encourage the students to look for ways to describe these relationships. It is important that students find the unknowns using mental calculation, as this will help them to more easily recognise the relationships.

  1. 1 – ? = 1
    2 – ? = 1
    3 – ? = 1
    4 – ? = 1
    456 - ? = 1

    Why is the right hand side of the equation always one?
    Use this pattern to solve:

    2000 - ? = 1
    1001 - ? = 2
  2. 0 + 1 + 2 = ?
    1 + 2 + 3 = ?
    2 + 3 + 4 = ?
    3 + 4 + 5 = ?
    Is there anything in common between the number in the ?’s?
    Why do you think this happens? (The number in the ? is three times the middle number on the left of the equation.)
    a + b + c = 300 , a, b and c are different numbers.
    What numbers could they be?
    What values for a, b and c would fit the pattern?
  3. 42 – 28 = ? – 30
    52 – 28 = ? – 30
    62 – 28 = ? - 30
    ...
    92 – 28 = ? - 30
    712 – 28 = ? – 30
    What rule can you find for equations in this pattern.
    How could this idea be used to solve 83 – 38?
  4. 1 x 9 + 1 = ?
    2 x 9 + 2 = ?
    3 x 9 + 3 = ?
    4 x 9 + 4 = ?
    ...
    In ? x 9 + ? = 100, what is ? if each ? is the same number?
    What rule can you find for all the equations in this pattern?
  5. 1 = ?, 11
    2 = ?, 11
    3 = ?, 11
    4 = ?, 11
    ...
    In ? = 77, 11, what is ?, ?
    In 33 = ?, 11, what is ?, ?
    What rule can you find for equations in this pattern?
    Would these equations be further down in the pattern? Give an explanation.? = 111, 11
    ? = 111 ÷ 11
    ? = 1034 ÷ 11

Session 3

Students solve “What’s my Number?” problems and record how they found the answer. At this stage trial and improvement are legitimate strategies though the problems are structured to encourage students to attend to structure and apply the processes of arithmetic.

  1. If you take my number, multiply it by three then add seven, you get fifty-two.
    What is my number? (15 since 15 x 3 + 7 = 52; or 3? + 7 = 52, so 3? = 45 and ? = 15.)
  2. If you take my number, subtract ten from it then divide it by two, you get sixteen.
    What is my number? (42 since 42 – 10 = 32 and 32 ÷ 2 = 16)
  3. If you take my number and add twenty-four to it, the answer is three times my number.
    What is my number? (X + 24 = 3X, so 24 = 2X and X = 12.)
  4. You take my number and divide it by three, the answer is the same as my number minus forty.
    What is my number? (60 since 60, 3 = 20 and 60 – 40 = 20.)
  5. If you take my number and multiply it by itself, the answer is my number added to itself.
    What is my number? (2 since 2 x 2 = 4 and 2 + 2 = 4 and 0 for the same reason; or ? x ? = ? + ? so ? x ? = 2?and ? = 0 or ? = 2.)

Session 4

This session involves students trying to work out the functional rule for given input and output numbers. These functions increase in complexity as the week progresses. Each example offers the students three input/output pairs. Below these three pairs are three other examples (typed in bold) that could be used if needed. On the card that is presented to students staple a small piece of card over the output of these last three pairs so that students can predict the output number then check by turning up the cover.

  1.  

In

5

2

7

4

1

3

Out

9

3

13

7

1

5

The rule is two times the input number less one gives the output number

  1.  

In

2

5

9

4

1

3

Out

7

16

28

13

4

10

The rule is times three plus one.

  1.  

In

8

3

6

4

1

5

Out

5

0

3

1

-2

2

The rule is take away three.

  1.  

In

10

2

7

4

1

8

Out

6

2

4.5

3

1.5

5

The rule is divide by two plus one.

  1.  

In

5

2

1

4

10

3

Out

25

4

1

16

100

9

The rule is the input number multiplied by itself (squared).

Session 5

These activities involve students in working out the number of counters or cubes that are in each cup of a given colour. Several clues are provided and students must combine these clues to find a solution. In each problem cover the top of the cups so students cannot look inside them. This is easily done by putting a screwed up piece of paper into the cup after the counters or cubes have been put in.
For each problem all students should solve it, recording their reasoning, before the solution is “revealed.”

  1. Into each yellow cup put four cubes, into each blue cup put five cubes.
    cups.
    Clues:
    14 cubes in total           16 cubes in total
     
  2. Into each yellow cup put three cubes, into each blue cup put six cubes.
    cups.
    Clues: Four yellow cups have the same number of cubes as two blue cups.
    cups.
    15 cubes in total
     
  3. Into each yellow cup put five cubes, into each blue cup put three cubes.
    cups.
    Clues: Three yellow cups have the same number of cubes as five blue cups.
    cups.
    11 cubes in total
     
  4. Into each yellow cup put two cubes, into each blue cup put four cubes, and into each red cup put six cubes.
    cups.
    Clues: Six yellow cups have the same number of cubes as three blue cubes that have the same number of cubes as two red cups.
    cups.
    14 cubes in total

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